COMPACT SETS WITHOUT CONVERGING SEQUENCES IN THE RANDOM REAL MODEL
A. DOW and D. FREMLIN
Abstract. It is shown that in the model obtained by adding any number of random reals to a model of CH, there is a compact Hausdorff space of weightω1 which contains no non-trivial converging sequences. It is shown that for certain spaces with no converging sequences, the addition of random reals will not add any converging sequences.
1. Introduction In this paper we will prove the following result.
Theorem 1.1. If we add any (cardinal) number of random reals to a model of ZFC + CH, we obtain a model in which there is a compact space of weightω1 with no non-trivial converging sequences.
It is well-known that in any model of Martin’s Axiom every compact space of weight less than the continuum is sequentially compact (see [6, 24G]).
One motivation for this result is an old question due to Efimov: Is there an infinite compact Hausdorff space without convergent sequences which does not include a copy ofβN?
Fedorchuk [5] gives a construction of such a space if the splitting number,s, isω1 and 2ω1 <2c; his method gives a space of weightcand cardinality 2ω1 (see [4] for generalizations to larger values ofs). Any model obtained by adding uncountably many random reals will be a model ofs=ω1. However our result provides new situations
Received July 18, 2005.
2000Mathematics Subject Classification. Primary 54A25; Secondary 03E35 54D30.
Key words and phrases. random reals, converging sequences, Efimov problem.
The authors thank the referee for a very careful review.
in which there is an Efimov space (not to mention that it has small weight) because in a model obtained by adding random reals one can arrange that 2ω1 does equal 2c. Talagrand [11] gives a construction dependent on the continuum hypothesis. Just and Koszmider [9] produce such examples in yet other models.
A spaceX isextremally disconnectedif disjoint open sets have disjoint closures (and the closure of every open set is open). A compact space X is anF-space if disjoint cozero sets have disjoint closures. The space βN is extremally disconnected whileβN\Nis anF-space that is not extremally disconnected. It is well known that an F-space does not have non-trivial converging sequences.
We will be using the technique of forcing. IfPis a poset in a modelV (of ZFC), and ifGis aP-generic filter (overV), thenV[G] is the forcing extension byP. IfKis a compact Hausdorff space in the modelV, then there is a natural and canonical compact Hausdorff space ˜K in V[G] definable fromK (up to homeomorphism). IfT is the topology onK, then it is well understood that inV[G], T forms a base for a topology onK, but that, in general, this very natural topology obtained fromK in V, is not going to be compact. For example if K is the unit interval, then ˜K is going to be the unit interval including the appropriately valued new reals. The spaceK with the topology generated by T will be a dense subset of ˜K. IfK is zero dimensional, then the most natural method to define ˜K is simply by Stone Duality: in V, CO(K) is the Boolean algebra of clopen sets and K is homeomorphic to the Stone space,S(CO(K)). We will let ˜K also be theS(CO(K)) as computed inV[G]. IfU is a closed subset ofK (inV), we will usually assume that ˜U is the natural closed subset of ˜K.
In the case thatK is not zero dimensional, we may assume thatK is embedded into [0,1]λ for some ordinal λ. When we pass toV[G], even [0,1] may be larger but we will still haveK sitting as a subset of [0,1]λ, and we simply take ˜K to be the closure ofK in [0,1]λ. More details on this idea can be found in [1] or [3, 5.1].
It is helpful to distinguish between βN in V[G] and βfN. If new reals are added to V[G], then βfN is not extremally disconnected because the base for the topology is the family{A˜:A⊂NandA∈V}.
Our main result is that if K is a compact F-space, then ˜K will not have non-trivial converging sequences.
The simplest example then is thatβfNwill have no non-trivial converging sequences. In the final section we give an example of a compact Hausdorff space K with no non-trivial converging sequences such that ˜K does have
non-trivial converging sequences after adding random reals. It follows easily from Koszmider’s results in [10] that in a Cohen real forcing extension, ˜K will have non-trivial converging sequences for every infinite compact space K.
2. No converging sequences in F-spaces
In the next section we derive Lemma2.1as a corollary to a main combinatorial lemma. In this section we establish the main Theorem1.1as a consequence of Lemma2.1.
Letκbe any cardinal and consider the probability measure µon the space 2κ in whichµ(bα) =µ(∼bα) = 12 for eachα∈κ, wherebαis the clopen set{f ∈2κ:f(α) = 0}and∼bα is the complement.
LetMdenote theσ-algebra generated by the basic clopen sets; thusMis the Baire sets. Random real forcing is the posetMwhich is obtained fromM \ {∅}by identifying two members ofMif the symmetric difference has measure 0 and ordered by inclusion (mod measure 0). The posetMis ccc and complete (every non-empty subset has a least upper bound).
In our treatment ofM, it will be useful to fix a choice of representatives fromM∪ {0}: for each element bof M, let hbi ∈ M be chosen from the equivalence class represented byb. For convenience, assume thath0iis the empty set andh1i is the entire measure space 2κ. We do not, however, make any other assumptions about this assignment, except of course, thata≤b∈Mwill imply thathai \ hbiwill have measure 0.
If ˙xis theM-name of an element of [0,1]λ(that is, 1x˙ ∈[0,1]λ), then for each basic open subsetU of [0,1]λ, there is a unique elementb∈M∪ {0}such that
bx˙ ∈U˜ and ∼bx /˙ ∈U˜
becauseMis complete. It is standard to let [[ ˙x∈U]] denote this elementb. Similarly, ifKis zero dimensional and U is a clopen subset ofK, we would also have that [[ ˙x∈U]] is exactly the condition b∈Msuch thatbx˙ ∈U˜ and∼bx /˙ ∈U˜. For improved readability, leth[ ˙x∈U]ibe an alternate notation forh[[ ˙x∈U]]i.
Lemma 2.1. Let K be a compact zero dimensional metric space, and let x,˙ {x˙i:i∈ω} be M-names of members ofK˜ such that
1x˙ ∈ {x˙i:i∈ω} and x˙i6= ˙xj(i < j).
If Y0 is any clopen subset ofK andX0∈ M is such thatX0 ⊂ h[ ˙x∈Y0]i, then for anyε >0, there is a clopen setU0 of Y0 such that
µ(X0\ [
i∈ω
h[ ˙xi∈U0]i)< ε and µ(h[ ˙x∈U0]i)< ε .
Proof. See Corollary3.2.
More informally, the idea is that with sufficiently high probability ˙xis not inU0 while at the same time, with sufficiently high probability at least one of the ˙xi’s is inU0.
The main result of this section is Lemma 2.2, we postpone its proof until we derive some corollaries and Theorem1.1.
Lemma 2.2. IfK is a compact zero dimensional space, andGisM-generic, then for each infinite setD⊂K,˜ there is a sequence,{Un :n∈ω} ∈V, of pairwise disjoint clopen subsets ofK, such that for eachn∈ω,D∩U˜n
is not empty.
Corollary 2.3. If K is a compact extremally disconnected space, and G is M-generic, then in V[G], every infinite subsetD ofK˜ has a subsetD0 such thatD0 maps continuously ontoβfN.
Proof. By Lemma2.2, there is a sequence {Um : m ∈ ω} of pairwise disjoint clopen sets with the property that, inV[G], ˜Um∩D is not empty for each m. Still inV, the mappingUm7→m, lifts to a mappingg fromK ontoβN. It is routine to check that there is a canonical reinterpretation ofginV[G] which maps ˜KontoβfNand sending ˜Umtom. It follows then that ifD0⊂D is any set such thatD0∩U˜mis not empty for eachm, then the
closure ofD0 maps continuously ontoβfN.
Clearly Theorem1.1 follows immediately from Corollary2.3. The following more general statement does not assume that the space is zero dimensional.
Corollary 2.4. If K is a compactF-space andG isM-generic then, inV[G], every infinite subset of K˜ has an infinite discrete subset whose closure maps onto βfN.
Proof. LetD be any countably infinite subset of ˜K inV[G]. By Corollary2.3, it suffices to show that there is a compact extremally disconnected subsetY of K(in V) such thatD⊂Y˜. By a standard technique in forcing, we can fix a countable set{x˙i:i∈ω}of M-names of elements of ˜K such that D={valG( ˙xi) :i∈ω}. Working in V, we have K embedded in [0,1]λ for some cardinalλ. LetI denote the ideal of cozero subsets U of [0,1]λ such that [[ ˙xi∈U]] = 0 for all i∈ω. Clearly ˜U∩D is empty for eachU ∈ I. LetY be the intersection of the family
{[0,1]λ\U :U ∈ I}
and notice thatD⊂Y˜ in V[G]. Every ccc closed subset of a compact F-space is anF-space, and consequently also extremally disconnected. We will be finished by showing thatY is ccc.
Let{Uα:α∈ω1}be any family of cozero subsets of [0,1]λsuch that{(Y∩Uα) :α∈ω1}are pairwise disjoint.
For eachα < β < ω1, we thus have that Uα∩Uβ ∈ I. It follows then that [[ ˙xi ∈Uα∩Uβ]] is 0, and therefore thath[ ˙xi∈Uα]i ∩ h[ ˙xi ∈Uβ]ihas measure 0. Therefore there is aδ < ω1 such that [[ ˙xi ∈Uα]] = 0 for allα≥δ andi∈ω. In other words,Uα∈ I for allα≥δ, proving that{Y ∩Uγ :γ∈ω1}is indeed countable.
Proof of Lemma2.2. Recall that we have thatK is a zero dimensional space,Gis anM-generic, and assume that D∈V[G] is an infinite subset of ˜K. Let{xi :i∈ω} be any infinite discrete subset ofD and letxbe any limit point. By standard forcing arguments, there is a name ˙x and a sequence of names{x˙i :i∈ ω} such that valG( ˙x) =xand valG( ˙xi) =xi for each i∈ ω. In addition, there is some element b∈ G, such that b x˙ is a limit of the infinite discrete set{x˙i :i∈ω} which is contained in ˜K. By a possible modification of the names ˙x and{x˙i:i∈ω}we can assume thatb is actually 1.
In order to apply Lemma2.1, we will have to replaceKby a suitable metric spaceY0(inV). To do so identify Kwith the Stone space of its clopen algebraCO(K). Fix for eachi∈ω, a name ˙Ciof a member ofCO(K) such that 1C˙i∩ {x˙j :j∈ω}={x˙i}. For eachi, there is a countable collectionCi⊂CO(K) such that [[U = ˙Ci]] = 0 for all U ∈CO(K)\ Ci. Let C be the countable subalgebra ofCO(K) that is generated by S
i∈ωCi. Let Y0 be the Stone space ofC. For eachi∈ω, there is a canonical name ˙yi for a member of ˜Y0 such that 1y˙i= ˙xi∩ C, and similarly a name ˙y such that 1y˙= ˙x∩ C. Clearly for eachU ∈ Cand eachi∈ω,valG( ˙xi)∈U˜ ⊂K˜ if and only ifvalG( ˙yi)∈U˜ ⊂Y˜0. Of course we are using ˜U in two different senses in the previous sentence.
To show that we have a sequence{Un :n ∈ω} ∈V, of pairwise disjoint clopen subsets of Y0 as required, it suffices to show that for any non-zero a∈ M, there is such a sequence {Un : n∈ ω} (depending on a) and an a0< asuch that, for alln,a0U˜n∩ {y˙i:i∈ω} 6=∅. Let a∈Mand chooseε >0 so thatε < µ(hai). We will be done if we now find a sequence{Um:m∈ω}so thatS
i<ωh[ ˙yi∈Um]ihas measure greater than 1−εfor allm.
This we do by repeated applications of Lemma 2.1. Set X0 = 2κ and ε0 = ε/2. By Lemma 2.1, there is a clopenU0⊂Y0such thatµ(h[ ˙y∈U0]i)< ε0andµ(X0\S
i∈ωh[ ˙yi ∈U0]i)< ε0. SetY1=Y0\U0,X1=h[ ˙y∈Y1]i, andε1=ε0/2. Observe thatX1 has measure greater than 1−ε0.
By induction onm, we select clopen sets Um of Ym and define Ym+1 = Ym\Um, and Xm+1 =Xm\ h[ ˙y ∈ Um]i = h[ ˙y ∈ Ym+1]i. The inductive assumptions are that µ(h[ ˙y ∈ Um]i) < ε/2m+1 and µ(Xm\S
i∈ωh[ ˙yi ∈ Um]i)< ε/2m+1. To see that this can continue, we need only observe that h[ ˙y∈Ym]ihas measure greater than 1−ε+ε/2msinceh[ ˙y∈S
n<mUm]ihas measure less than P
n<mε/2n+1. Since Xm has measure greater than 1−ε+ε/2m and µ(Xm\S
i∈ωh[ ˙yi ∈ Um]i) < ε/2m+1, it follows that S
i∈ωh[ ˙yi∈Um]ihas measure greater than 1−ε. This completes the proof.
3. The main combinatorial lemma
In the case that K is a compact metric space and ˙xis the M-name of a member of ˜K, a common approach to random real forcing would be to construct a Borel measurable functionf from the measure space (2κ,M, µ) into K so that for each open setU ⊂K,h[ ˙x∈U]i=f−1(U) (mod measure 0).
If ˙xand ˙yare each names of members of ˜K, then 1x˙ 6= ˙y is equivalent to the following statement: there is a countable setB⊂Msuch thatS
b∈Bhbihas measure 1, and for eachb∈B, there are disjoint basic openU, Wsuch that
bx˙ ∈U˜ andby˙ ∈W˜. This is the same as asserting that b≤[[ ˙x∈U]]∩[[ ˙y∈W]].
Thus, if {x˙i : i ∈ ω} is a sequence of M-names of members of ˜K, then for each i, we would have a Borel measurable function fi corresponding to ˙xi. If we assume that, for i < j, 1 x˙i 6= ˙xj, then we have that {x ∈ 2κ : fi(x) = fj(x)} has measure 0. This translation helps explain the interest in the following main combinatorial Lemma. For the other results in this paper, the simplified version in Corollary3.2is sufficient and its proof can be read independently. The lemma derives from Shelah via 1G of [8] and [2].
Lemma 3.1. Let (X,Σ, µ) be a probability space and Y a separable metric space. Suppose we have a non- negative finitely additive functional ν defined on the Borel subsets of Y, with νY = 1, and a sequence hfiii∈N of measurable functions, each defined on a set Xi ∈ Σ and taking values in Y, such that {x : x ∈ Xi∩Xj, fi(x) =fj(x)} is negligible wheneveri6=j.
Set Z = T
j∈N
S
i≥jXi. Then for any ε > 0 there is an open set H ⊂ Y such that νH ≤ ε and µ(Z \ S
i∈Nfi−1[H])≤ε.
Proof. (a) Setη= 14ε >0 and letM ∈Nbe such that (1−η)M < η. LetN ∈Nbe such thatµ(Z\Z0)≤η, where
Z0={x∈Z,|{i:i < N, x∈Xi}| ≥M}.
(b) Let A be the ideal of subsets A ⊂ Y such that inf{νG : A ⊆ G, Gopen in Y}
= 0. Let E be the algebra of {E : E ⊆ Y, ∂E ∈ A} where ∂E = E \intE is the boundary of E. Then there is a finite partitionDofY into members ofE such thatµB≤η, where
B= [
i<j<N
[
D∈D
fi−1[D]∩fj−1[D].
Proof of (b): For any given metric onY, letB(y, α) denote the open ball of radius αcentered aty. Then for anyy0∈Y, the mapping sending a positive realαtoνB(y0, α) is non-decreasing, therefore continuous at all but countably many points. This means thatB(y0, α)∈Efor all but countably manyα, hence the open sets inEis a base forY. There is therefore a sequencehUnin∈N inE running over a base forY. For each n, let Dn be the partition ofY generated by{Ui:i≤n}, and set
Bn= [
i<j<N
[
D∈Dn
fi−1[D]∩fj−1[D].
ThenhBnin∈N is non-increasing and
\
n∈N
Bn= [
i<j<N
{x:x∈Xi∩Xj, fi(x) =fj(x)}
is negligible, so there is somensuch thatµBn≤η, and we can takeD=Dn. This completes the proof of (b).
(c) EnumerateD as {Er :r < m}. Let K ⊂mbe a random set, in which eachr < m has independently, a probabilityη of appearing.
That is, we are defining a counting measureρonP(m) by
ρ({K}) =η|K|(1−η)m−|K|=η|K|(1−η)|m\K|.
SetFK =S
r∈KEr for eachK. Then for any particularx∈Z0\B, Pr(x /∈fi−1[FK] for every i < N) = Pr(Kx∩K=∅) (where Kx={r:x∈fi−1[Er] for some i < N})
= (1−η)|Kx|≤(1−η)M ≤η because ifx∈Z0\B then
|Kx|=|{i:i < N , x∈Xi}| ≥M.
So
Pr µ (Z0\B)\ [
i<N
fi−1[FK]
!
≤2ηµ(Z0\B)
!
≥ 1 2.
A reader with less experience with probability may prefer more basic explanations. Consider the product measure spaceP(m)×(Z0\B). Define the subset S by
S = [
K⊂m
{K} × [
i<N
fi−1[[
r∈K
Er]∩(Z0\B)
! .
For each K ⊂ m, let SK denote the vertical section S
i<Nfi−1[S
r∈KEr] and for each x∈ (Z0\B), we let Sx denote the horizontal section atx.
It is easily verified that for eachx∈(Z0\B),Sx={K∈ P(m) :K∩Kx6=∅}; henceρSx>1−η. From this it follows that (ρ×µ)(S)>(1−η)µ(Z0\B).
The statement that
Pr µ (Z0\B)∩ [
i<N
fi−1[[
r∈K
Er]
!
≥(1−2η)µ(Z0\B)
!
≥ 1 2
is just the statement that
T ={K:µ((Z0\B)∩ [
i<N
fi−1[[
r∈K
Er])≥(1−2η)µ(Z0\B)}
={K:µ(SK)≥(1−2η)µ(Z0\B)}
has measure greater than 12. To show this observe that (1−η)µ(Z0\B)<(ρ×µ)(S)
≤(ρ×µ)(S∩(T×(Z0\B)) + (ρ×µ)(S∩(∼T×(Z0\B))
≤ρ(T)µ(Z0\B) +ρ(∼T)(1−2η)µ(Z0\B).
Therefore
1−η≤ρ(T) +ρ(∼T)−2ηρ(∼T) = 1−2ηρ(∼T), from which it follows thatρ(∼T)≤ 12.
At the same time,
E(νFK) =
m−1
X
r=0
νEr Pr(r∈K) =η, so Pr(νFK >2η)<12, and there must be someK such that
νFK≤2η, µ((Z0\B)\ [
i<N
fi−1[FK])≤2ην(Z0\B).
But this means that
µ(Z\ [
i∈N
fi−1[FK])≤2η+µB+µ(Z\Z0)≤ε.
A similar product measure argument usingρ×ν onP(m)×Y can be used for more detailed explanations in
this case as well.
Recall we are assuming that we have a sequence{x˙i :i∈ω} ofM-names of members of ˜K for some compact metricK. We will simplify the discussion somewhat and assume that K is also zero dimensional. Let us also assume that 1{x˙i :i∈ω} is discrete (and of course that 1x˙i6= ˙xj fori6=j). Since 1K˜ is compact, we may fix a name ˙xof a member of ˜K such that 1x˙ is a limit point of {x˙i:i∈ω}(and notice that 1x˙ 6= ˙xi
for eachi∈ω).
The following result is really a corollary to the main lemma but for greater clarity for most readers, we will reprove it based on the ideas in the main lemma but using more traditional forcing notation and a direct approach to inductively choose clopen setsDto union up to the desired clopen setU0. To apply Lemma3.1, one could set ν(U) to beµh[ ˙x∈U]ifor each openU ⊂K.
Corollary 3.2. Let K be a compact zero dimensional metric space, and let x,˙ {x˙i : i ∈ ω} be M-names of members ofK˜ such that
1x˙ ∈ {x˙i:i∈ω} and x˙i6= ˙xj(i < j).
If Y0 is any clopen subset ofK andX0∈ M is such thatX0 ⊂ h[ ˙x∈Y0]i, then for anyε >0, there is a clopen setU0 of Y0 such that
µ(X0\ [
i∈ω
h[ ˙xi∈U0]i)< ε and µ(h[ ˙x∈U0]i)< ε .
Proof. Let 0< η < ε/5, thus we can afford to throw away 4 pieces ofX0of size less thanη. Let{an:n∈ω}
be a clopen basis forY0. Fix an integerLso large that 4/√ L < η.
Since X0 ⊆ h[ ˙x∈ Y0]i and 1 x˙ ∈ {x˙i :i∈ω}, X0\S
i∈ω\mh[ ˙xi ∈ Y0]i has measure 0 for each integer m.
Therefore, for eachm, there is anm0> msuch thatX0\S
i∈[m,m0)h[ ˙xi∈Y0]ihas measure less thanη/L. Fix an increasing sequencem0< m1<· · · such thatX0\S
i∈[mj,mj+1)h[ ˙xi∈Y0]ihas measure less thanη/Lfor eachj.
LetM =mL, and setX00 =T
j<L
S
i∈[mj,mj+1)h[ ˙xi∈Y0]i. By construction,µ(X0\X00)< ηand for eachx∈X00, {i < M :x∈ h[ ˙xi∈Y0]i}has cardinality at leastL.
LetY0={y∈Y0: [[ ˙x=y]]>0}. Since [[ ˙x=y]]∧[[ ˙x=y0]] = 0 fory 6=y0, it follows that there is a finite set F ⊂Y0 such thatS
y∈Y0\Fh[ ˙x=y]ihas measure less thanη/2.
For eachy∈F, there is any such thaty∈any andµ(h[ ˙xi∈any]i)< η/(2M· |F|) for eachi < M. Therefore S
i<M,y∈Fh[ ˙xi∈any]ihas measure less than η/2. Letj0 be larger thanny for eachy∈F. Set X1=X00 \ ( [
i<M,y∈F
h[ ˙xi∈any]i ∪ [
y∈Y0\F
h[ ˙x=y]i).
For eachj∈ω, letDj be the partition ofY0generated by{an :n < j}. That is, for eachD∈ Djand eachn < j, eitherD⊂an orD∩an is empty. For eachj andi < i0< M, letE(j, i, i0) =S
D∈Djh[ ˙xi∈D]i ∩ h[ ˙xi0 ∈D]i. Let E(ω, i, i0) =T
j∈ωE(j, i, i0).
Since 1 x˙i 6= ˙xi0, we have that E(ω, i, i0) has measure 0. Also, we have thatE(j+ 1, i, i0)⊂E(j, i, i0) for eachj. Therefore, there is some j such thatE(j, i, i0) has measure less thanη/M2. It follows that there is an integerj > j0 such thatE(j, i, i0) has measure less thanη/M2for eachi < i0< M. LetD=Dj.
SetX2=X1\S
i<i0<ME(j, i, i0) and again we have thatX1\X2has measure less thanη. Now it follows that for eachx∈X2,Dx={D∈ D: (∃i < M)x∈ h[ ˙xi∈D]i}has cardinality at leastLand thatDx∩DF =∅whereDF = {D∈ D:D∩F 6=∅}.
LetJ be an integer so large that 1/J < µ(h[ ˙x∈D]i ∩X2) for eachD∈ D \ DF.
Consider anyD∈ D \ DF and y∈D. Since y /∈F, h[ ˙x=y]i ∩X2 has measure 0, hence there is a clopen set ay⊂D such thath[ ˙x∈ay]i ∩X2 has measure less than 1/(4J). Therefore there is a finite cover of Dby clopen setsE each with the property thath[ ˙x∈E]i ∩X2 has measure less than 1/(4J). Therefore there is a partition ED ofD by sets each of which has measure greater than 1/J and less than 5/(4J). By refiningD \ DF, we may assume that forD∈ D \ DF,h[ ˙x∈D]i ∩X2 has measure greater than 1/J and less than 5/(4J). It also follows that|D \ DF|< J.
For eachD∈ D, letν(D) =µ(h[ ˙x∈D]i ∩X2). Of course, more generally we could defineν(E) =µ(X2∩ h[ ˙x∈ E]i) for all clopen subsets ofY but we only extend it to the simple finite algebra (henceσ-algebra) generated by D. Then consider the product measure (µ×ν) onX2×Y0.
Define the setS⊂X2×Y0 by
S = [
D∈D
[
i<M
(X2∩ h[ ˙xi∈D]i)×D and for eachx∈X2, we have the vertical fiberSx which is equal to
[Dx=[
{D∈ D: (∃i < M)x∈ h[ ˙xi∈D]i}.
Since|Dx| ≥L, we have thatν(Sx)>(L/J). Therefore
(µ×ν)(S)>(L/J)µ(X2).
For eachD∈ D,
SD= [
i<M
X2∩ h[ ˙xi∈D]i
and
(L/J)µ(X2)<(µ×ν)(S) = X
D∈D\DF
ν(D)µ(SD)< J· 5 4J ·max
D∈D(µ(SD)).
Since (L/J)µ(X2)< J·(5/(4J))·max(µ(SD)), we have 4
5·L
Jµ(X2)<max
D∈D(µ(SD)).
Thus there is aD0∈ D \ DF such thatµ(SD0)>(4L/5J)µ(X2).
Now setX3=X2\(h[ ˙x∈D0]i ∪SD0) andY3=Y0\D0. SetS3=S∩(X3×Y3).
Notice that forx∈X3,D0∈ D/ x, hence we still have thatν((S3)x)> L/J. Ifµ(X3)≤1/√
Lwe stop, otherwise, arguing exactly as above, there is aD1∈ D \({D0} ∪ DF), such thatµ(SD1∩X3)>(4L/5J)µ(X3).
Inductively we keep choosing suchDm∈ D \(DF∪{D0, . . . , Dm−1}), such thatµ(SDm∩X2+m)>(4/5)·(L/J)·
µ(X2+m) and set X2+m+1=X2+m\(Dm∪SDm). Let` be the first valuem+ 1 such thatµ(X2+m+1)≤1/√ L.
Now we consider the families of disjoint sets{SDm ∩X2+m:m < `} and the fact that X2\X2+` is covered by this union together with the union of the family{h[ ˙x∈Dm]i:m < `}.
For each m < `, µ(SDm ∩X2+m) is bigger than (4/5)·(L/J)µ(X2+m)> (4/5)·(L/J)·(1/√
L). Therefore, the measure of the union of the family{SDm∩X2+m:m≤`}is bigger than`·(4L/5J)·(1/√
L). The union of the familyh[ ˙x∈Dm]i(m < `) has some measurepwhich is less than (5/4)·`·(1/J). Therefore the measure of T =S{SDm∩X2+m:m < `} is more than (16/25)·√
L·p.
Now, 1>(16/25)·√
L·pand (25/16)·(1/√ L)> p.
Also, sinceX2\X` is covered by these sets, we haveµ(T) +p > µ(X2)−1/√
L; or µ(T)> µ(X2)−η/4−p >
µ(X2)−η/4−(25/16)·(1/√
L) which in turn is greater thanµ(X2)−η/4−(25/16)·(η/4)> µ(X2)−(3/4)η. It follows thatµ(T) is greater thanµ(X0)−εas required. So we setU0=D0∪ · · · ∪D`−1.
Finally we computeµ(h[ ˙x∈U0]i), it equalsµ(h[ ˙x∈S
m<`Dm]i) =p <(25/16)·(1/√
L)<(25/16)(η/4)< ε.
4. An example that acquires converging sequences
Proposition 4.1. There is an example of a compact Hausdorff space K which has no non-trivial converging sequences such thatK˜ does have non-trivial converging sequences after adding random reals.
Proof. Recall thatM∪{0}(with measureµ) is the random real forcing algebra obtained from the measure space X = 2κfor some infinite cardinalκ. LetZ denote the Stone space ofM∪ {0}. We will define a compactification KofZ×N. K will be the Stone space of a Boolean subalgebraB of the natural algebraN= (M∪ {0})Nwhere
B={hbnin∈N: min (Σn∈N µ(bn),Σn∈Nµ(X\bn)) is finite}.
Now
{hbnin∈N:P
n∈Nµ(X\bn) is finite}
is an ultrafilter inB with limitz inK.
It is fairly routine to prove that K contains no non-trivial converging sequences (much like proving that S(M∪ {0}) contains none).
Letν denote the usual product measure onN. We will now force with the random real poset obtained fromN with the measureν. LetGbe a generic filter and for eachn, letgn∈Z˜× {n} be determined by the (ultra)filter {bn:h1,1, . . . ,1, bn,1,1, . . .i ∈G}. We show that{gn:n∈N} converges toz. Assume thathbnin∈
N∈B is such that z /∈˜b and suppose that ais in the forcing poset. We show that there is an extension (subset) of awhich forces that ˜b∩ {gn : n∈N} is finite. Let 0< ε be less thanν(hai). Let, for n∈N, bn =h1,1, . . . ,1, bn,1, . . .i in the measure algebraN. Observe thatν(bn) =µ(hbni). There is an integerM such thatP
n>Mµ(hbni) is less thanε, hence ν(W
n>Mbn)< ε. This proves that ν(a\W
n>Mbn)>0. The conditiona0 =a\W
n>Mbn forces
thatgn ∈/˜bfor eachn > M.
1. Balogh Z. and Gruenhage G.,Two more perfectly normal non-metrizable manifolds, Topology Appl.151(1–3) (2005), 260–272, MR 2139756.
2. Burke M. R.,A theorem of Gitik and Shelah on disoint refinements of sequences of subsets of the real line, Note of 23.8.96.
3. Dow A.,Set theory in topology, Recent progress in general topology (Prague, 1991), North-Holland, Amsterdam, 1992, 167–197, MR 1229125.
4. ,Efimov spaces and the splitting number, Topology Proceedings, to appear.
5. Fedorchuk V. V.,A compact space having the cardinality of the continuum with no converging sequences, Math. Proc. Cambridge Phil. Soc.81(1977), 177–181.
6. Fremlin D. H.,Consequences of Martin’s axiom, Cambridge Tracts in Mathematics, vol. 84, Cambridge University Press, Cam- bridge, 1984, MR780933 (86i:03001).
7. ,Real-valued-measurable cardinals, Proceedings of the Bar-Ilan Conference on Set Theory of the Reals 1991, Amer. Math.
Soc. (Israel Mathematics Conference Proceedings 6), 1993, 151–304.
8. ,A theorem of Gitik and Shelah on disjoint refinements of sequences of subsets of the real line, Note of 4. 4. 97.
9. Just W. and Koszmider P.,Remarks on Cofinalities and Homomorphism Types of Boolean Algebras, Algebra Universalis 28 (1991), 138–149.
10. Koszmider P.,The Consistency of the negation of CH and the pseudoaltitude less or equal to omega one, Algebra Universalis 27(1990), 80–87.
11. M. Talagrand,Un nouveauC(K)qui poss`ede la propri´et´e de Grothendieck, Israel J. Math.37(1980) 181–191.
A. Dow, University of North Carolina Charlotte,e-mail:[email protected] D. Fremlin, University of Essex,e-mail:[email protected]